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Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions

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arXiv 2023年

作者： Lin, Bin Mao, Zhiping Wang, Zhicheng Karniadakis, George Em School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China Laboratory of Ocean Energy Utilization Ministry of Education School of Energy and Power Engineering Dalian University of Technology Dalian China Division of Applied Mathematics Brown University ProvidenceRI02906 United States

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has been introduced and extensively employed in approximating solution of PDEs. Nevertheless, to solve problems consisting of sharp solutions poses a significant challenge when employing these two approaches. To address this issue, we propose in this work a novel framework termed Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize DeepONet to learn the solution operator for a set of smooth problems relevant to the PDEs characterized by sharp solutions. Subsequently, we integrate the pre-trained DeepONet with PINN to resolve the target sharp solution problem. We showcase the efficacy of OL-PINN by successfully addressing various problems, such as the nonlinear diffusion-reaction equation, the Burgers equation and the incompressible Navier-Stokes equation at high Reynolds number. Compared with the vanilla PINN, the proposed method requires only a small number of residual points to achieve a strong generalization capability. Moreover, it substantially enhances accuracy, while also ensuring a robust training process. Furthermore, OL-PINN inherits the advantage of PINN for solving inverse problems. To this end, we apply the OL-PINN approach for solving problems with only partial boundary conditions, which usually cannot be solved by the classical numerical methods, showing its capacity in solving ill-posed problems and consequently more complex inverse problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Partial differential equations

Evidence-Aware Multi-Modal Data Fusion and its Application to Total Knee Replacement Prediction

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Evidence-Aware Multi-Modal Data Fusion and its Application t...

Proceedings of the Digital Image computing: Technqiues and Applications (DICTA)

作者： Xinwen Liu Jing Wang S. Kevin Zhou Craig Engstrom Shekhar S. Chandra School of Electrical Engineering and Computer Science The University of Queensland Brisbane Australia The Commonwealth Scientific and Industrial Research Organisation Canberra Australia Center for Medical Imaging Robotics Analytic Computing & Learning (MIRACLE) School of Biomedical Engineering & Suzhou Institute for Advanced Research University of Science and Technology of China Suzhou China Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS) Institute of Computing Technology CAS Beijing China School of Human Movement and Nutrition Sciences The University of Queensland Brisbane Australia

ISBN: (数字)9798350379037

ISBN: (纸本)9798350379044

Deep neural networks have been widely studied to predict a medical condition, such as total knee replacement (TKR). It has shown that data of different modalities, such as imaging data, clinical variables, and demographic information, provide complementary information and thus can improve the prediction accuracy together. However, the data sources of various modalities may not always be of high quality, and each modality may have only partial information of medical condition. Thus, predictions from different modalities can be in conflict, and the final prediction may fail in the presence of such a conflict. Therefore, it is important to account for the reliability of each source data and the prediction output when making a final decision. In this paper, we propose an evidence-aware multimodal data fusion framework based on the Dempster-Shafer theory (DST). The backbone models contain an image branch, a non-image branch and a fusion branch. For each branch, there is an evidence network that takes the extracted features as input and outputs an evidence score, which is designed to represent the reliability of the output from the current branch. The output probabilities along with the evidence scores from multiple branches are combined with the Dempster's combination rule to make a final prediction. Experimental results on the public OA initiative (OAI) dataset for the TKR prediction task show that the proposed method has better performance by accounting for conflicts from various modalities.

关键词： Medical conditions Soft sensors Digital images Data integration Imaging Artificial neural networks Reliability theory Feature extraction Reliability engineering Knee replacement

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

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Physical Review E 2021年第3期104卷 035308-035308页

作者： Xinmeng Chen Zhenhua Chai Jinlong Shang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

关键词： Lattice-Boltzmann methods

Stochastic Gauss-Newton Algorithms for Online PCA

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arXiv 2022年

作者： Zhou, Siyun Liu, Xin Xu, Liwei School of Mathematical Sciences University of Electronic Science and Technology of China China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China

In this paper, we propose a stochastic Gauss-Newton (SGN) algorithm to study the online principal component analysis (OPCA) problem, which is formulated by using the symmetric low-rank product (SLRP) model for dominant eigenspace calculation. Compared with existing OPCA solvers, SGN is of improved robustness with respect to the varying input data and algorithm parameters. In addition, turning to an evaluation of data stream based on approximated objective functions, we develop a new adaptive stepsize strategy for SGN (AdaSGN) which requires no priori knowledge of the input data, and numerically illustrate its comparable performance with SGN adopting the manaully-tuned diminishing stepsize. Without assuming the eigengap to be positive, we also establish the global and optimal convergence rate of SGN with the specified stepsize using the diffusion approximation *** Codes 65F15, 68W27, 60H10, 60J70 Copyright © 2022, The Authors. All rights reserved.

关键词： Principal component analysis

A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

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Communications on Applied Mathematics and Computation 2020年第2期2卷 179-199页

作者： Junying Cao Ziqiang Wang Chuanju Xu School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

关键词： Caputo-Fabrizio derivative Fractional diferential equations High-order numerical scheme

A ternary phase-field model for two-phase flows in complex geometries

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arXiv 2023年

作者： Zhan, Chengjie Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this work, a ternary phase-field model for two-phase flows in complex geometries is proposed. In this model, one of the three components in the classical ternary Cahn-Hilliard model is considered as the solid phase, and only one Cahn-Hilliard equation with degenerate mobility needs to be solved due to the condition of volume conservation, which is consistent with the standard phase-field model with a single-scalar variable for two-phase flows. To depict different wetting properties at the complex fluid-solid boundaries, the spreading parameters in ternary phase-field model are determined based on the Young's law, in which the liquid-solid surface tension coefficient is assumed to be a linear function of gas-liquid surface tension coefficient and related to the contact angle and the minimum curvature of the solid surface. In addition, to achieve a high viscosity in the solid phase and preserve the velocity boundary conditions on the solid surface, the phase-field variable of the solid phase is also used to derive the modified Navier-Stokes equations. To test the present model, we further develop a consistent and conservative Hermite-moment based lattice Boltzmann method where an adjustable scale factor is introduced to improve the numerical stability, and conduct the numerical simulations of several benchmark problems. The results illustrate that present model has the good capability in the study of the two-phase flows in complex geometries. © 2023, CC BY.

关键词： Two phase flow

Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach

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arXiv 2024年

作者： Jin, Shi Liu, Nana Wei, Wei Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China Shanghai Artificial Intelligence Laboratory Shanghai China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China

Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems. To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic differential equations, utilizing the Schrödingerisation method for the corresponding approximate equation by treating the noise term as a (discrete-in-time) forcing term. Our algorithms are applicable to stochastic differential equations with both Gaussian noise and α-stable Lévy noise. The gate complexity of our algorithms exhibits an O(dlog(Nd)) dependence on the dimensions d and sample sizes N, where its corresponding classical counterpart requires nearly exponentially larger complexity in scenarios involving large sample sizes. In the Gaussian noise case, we show the strong convergence of first order in the mean square norm for the approximate equations. The algorithms are numerically verified for the Ornstein–Uhlenbeck processes, geometric Brownian motions, and one-dimensional Lévy flights. © 2024, CC BY.

关键词： Quantum computers

Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows

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Physical Review E 2022年第2期106卷 025319-025319页

作者： Chengjie Zhan Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

关键词： Contact line dynamics Multiphase flows Rayleigh-Taylor instability

Asynchronous Activity Detection for Cell-Free Massive MIMO: From Centralized to Distributed Algorithms

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arXiv 2022年

作者： Li, Yang Lin, Qingfeng Liu, Ya-Feng Ai, Bo Wu, Yik-Chung Shenzhen Research Institute of Big Data Shenzhen518172 China State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing100044 China The Department of Electrical and Electronic Engineering The University of Hong Kong Hong Kong The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Peng Cheng Laboratory Shenzhen518055 China Henan Joint International Research Laboratory of Intelligent Networking and Data Analysis Zhengzhou University Zhengzhou450001 China

Device activity detection in the emerging cell-free massive multiple-input multiple-output (MIMO) systems has been recognized as a crucial task in machine-type communications, in which multiple access points (APs) jointly identify the active devices from a large number of potential devices based on the received signals. Most of the existing works addressing this problem rely on the impractical assumption that different active devices transmit signals synchronously. However, in practice, synchronization cannot be guaranteed due to the low-cost oscillators, which brings additional discontinuous and nonconvex constraints to the detection problem. To address this challenge, this paper reveals an equivalent reformulation to the asynchronous activity detection problem, which facilitates the development of a centralized algorithm and a distributed algorithm that satisfy the highly nonconvex constraints in a gentle fashion as the iteration number increases, so that the sequence generated by the proposed algorithms can get around bad stationary points. To reduce the capacity requirements of the fronthauls, we further design a communication-efficient accelerated distributed algorithm. Simulation results demonstrate that the proposed centralized and distributed algorithms outperform state-of-the-art approaches, and the proposed accelerated distributed algorithm achieves a close detection performance to that of the centralized algorithm but with a much smaller number of bits to be transmitted on the fronthaul links. Copyright © 2022, The Authors. All rights reserved.

关键词： Internet of things